A question posed by Li in 2011 asks for the classification of transcendental meromorphic solutions with few poles (that is, with N(r,f)=S(r,f)N(r,f)=S(r,f)) of the nonlinear complex differential equation fn+Pd(z,f)=p1eα1(z)+p2eα2(z)f^n + P_d(z,f) = p_1e^{\alpha_1(z)} + p_2e^{\alpha_2(z)}, where PdP_d is a differential polynomial in ff of total degree dd, in the critical case d=n−1d = n-1 with constant nonzero p1,p2p_1, p_2 and distinct non-constant polynomial exponents. For d≤n−2d \le n-2 the solutions are known to be short exponential sums; the case d=n−1d = n-1 is open.

Background

Nevanlinna theory describes the value distribution of meromorphic functions through the characteristic function T(r,f)T(r,f) together with the proximity term m(r,⋅)m(r,\cdot) and the counting term N(r,⋅)N(r,\cdot); a quantity is denoted S(r,f)S(r,f) if it is o(T(r,f))o(T(r,f)) outside a set of finite logarithmic measure, and a meromorphic function aa is called a small function of ff if T(r,a)=S(r,f)T(r,a)=S(r,f). A classical line of research on the value distribution of differential polynomials studies which nonlinear complex ordinary differential equations with exponential-sum right-hand sides admit transcendental meromorphic solutions and what form those solutions must take. The central equation family is

fn+Pd(z,f)=p1(z)eα1(z)+p2(z)eα2(z),f^n + P_d(z,f) = p_1(z)e^{\alpha_1(z)} + p_2(z)e^{\alpha_2(z)},

where n≥2n\ge 2 is an integer, Pd(z,f)=∑λ∈Λaλ(z)∏i≥0f(i)(z)λiP_d(z,f)=\sum_{\lambda\in\Lambda}a_\lambda(z)\prod_{i\ge 0} f^{(i)}(z)^{\lambda_i} is a differential polynomial in ff and its derivatives with rational-function coefficients and total degree d=max⁡λ∑iλid=\max_\lambda\sum_i\lambda_i, p1,p2p_1,p_2 are nonzero rational functions, and α1,α2\alpha_1,\alpha_2 are distinct non-constant polynomials. Write Γ0={λ0eλz}\Gamma_0=\{\lambda_0 e^{\lambda z}\}, Γ1={λ1eλz+λ0}\Gamma_1=\{\lambda_1e^{\lambda z}+\lambda_0\}, Γ2={λ1eλz+λ2e−λz}\Gamma_2=\{\lambda_1e^{\lambda z}+\lambda_2e^{-\lambda z}\} and Γ3={λ1eλz+λ2e−2λz}\Gamma_3=\{\lambda_1e^{\lambda z}+\lambda_2e^{-2\lambda z}\}, with nonzero constants λ,λj\lambda,\lambda_j. Under the pole condition N(r,f)=S(r,f)N(r,f)=S(r,f) (finitely many poles up to a negligible remainder), the known classifications confine solutions to short exponential forms built from these classes. Li [Li 2011] proved that for d≤n−2d\le n-2, constant p1,p2p_1,p_2 and linear exponents αjz\alpha_j z with α1≠α2\alpha_1\ne\alpha_2, every transcendental meromorphic solution with N(r,f)=S(r,f)N(r,f)=S(r,f) has one of the forms c0+c1eα1z/nc_0+c_1e^{\alpha_1z/n}, c0+c2eα2z/nc_0+c_2e^{\alpha_2z/n}, or c1eα1z/n+c2eα2z/nc_1e^{\alpha_1z/n}+c_2e^{\alpha_2z/n} with α1+α2=0\alpha_1+\alpha_2=0, where c0c_0 is a small function of ff and the constants satisfy c1n=p1c_1^n=p_1, c2n=p2c_2^n=p_2; Liao, Yang and Zhang [Liao et al. 2013] extended the classification to nonzero rational pjp_j and general non-constant polynomial exponents at the same degree bound d≤n−2d\le n-2, concluding that solutions are rational-function multiples q(z)eP(z)q(z)e^{P(z)} of a single polynomial exponential, two-term sums c1(z)eβ(z)+c2(z)e−β(z)c_1(z)e^{\beta(z)}+c_2(z)e^{-\beta(z)}, or solutions of associated first-order linear differential equations. The critical case d=n−1d=n-1, in which the differential polynomial sits one degree below the pure power, is precisely where the degree comparison between fnf^n and PdP_d degenerates and the standard arguments fail: Li posed this case as an open question, and it remains unanswered without additional hypotheses on the exponents.

Problem Statement

Classify, without any restrictions beyond those stated, all transcendental meromorphic solutions ff with N(r,f)=S(r,f)N(r,f)=S(r,f) of

fn+Pd(z,f)=p1eα1(z)+p2eα2(z)f^n + P_d(z,f) = p_1e^{\alpha_1(z)} + p_2e^{\alpha_2(z)}

in the critical case d=n−1d=n-1: n≥2n\ge 2 is an integer; Pd(z,f)P_d(z,f) is a differential polynomial in ff with rational-function coefficients and total degree exactly n−1n-1; p1,p2p_1,p_2 are nonzero constants (Li's original setting; treating nonzero rational pj(z)p_j(z) simultaneously is welcome); α1,α2\alpha_1,\alpha_2 are distinct non-constant polynomials. The classification should determine (i) whether every such solution is necessarily a short exponential form of the kind that prevails for d≤n−2d\le n-2 — a rational-function multiple of one or two polynomial exponentials, as in the Γ\Gamma classes and their rational-coefficient analogues — or whether genuinely new forms appear at d=n−1d=n-1; (ii) the exact finite list of admissible forms together with the constraints among the constants (analogous to c1n=p1c_1^n=p_1, c2n=p2c_2^n=p_2, α1+α2=0\alpha_1+\alpha_2=0 at d≤n−2d\le n-2); and (iii) that every solution has one of the listed forms, together with data realizing each listed form. The question asks for the unrestricted general case, with no additional hypotheses on the exponents or the solution class — no restriction on the ratio α1′/α2′\alpha_1'/\alpha_2', not constant exponents only, and not entire solutions only.

References

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